College Master - Math Problem Example

College Math Written by Bipin Musale We first try to solve the first problem. In one rotation of the wheel the distance traveled by it when it rolls along a road is equal to the circumference of the wheel. In other words when the wheel travels a distance equal to its circumference it completes one rotation.Therefore, the number of rotations when the wheel with diameter of 15 inch travels a distance of 2000 miles is approximately equal to (2000 X 1760 X 3 X 12) / (3.14 X 15) i.e. 2,690,446.The number of rotations when the wheel with diameter of 16 inch travels a distance of 2000 miles is approximately equal to (2000 X 1760 X 3 X 12) / (3.14 X 16) i.e. 2,522,293.

Since the number of rotations for the wheel with diameter 15 inch are more its tires are more likely to wear out first and hence would need to be replaced earlier.Here it is given that the tires are equally durable(i.e. identical in terms of quality and thickness). Also we assume here that the roads on which the two cars travel and other conditions are identical so that the friction offered is practically the same and that the cars have more or less the same weight(generally the cars with the larger wheels have more weight and hence offer more burden on their wheels).

The difference in their weight, if any, is assumed to be insignificant enough not to contribute any significant difference in wearing out of the tires.Thus the car with the smaller wheels(diameter 15 inch) is likely to have its tires replaced first.Now, we take up the first part of the second problem. We know from trigonometry tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x).tan(y)) which is the tangent sum formula.Therefore, tan(x + 450) = (tan(x) + tan(450)) / (1 - tan(x).tan(450)).We also know that tan(90) is undefined and so also tan((4n+1).90) where n is any positive integer.

Therefore, tan(450) = tan((4X1+1)90) is undefined. Therefore, the right hand side of the above equation will be undefined and hence tan(x + 450) cannot be simplified using the tangent sum formula.But sin(x + 450) = cos(x) and cos(450 + x) = -sin(x) as x + 450 is located in second quadrant. Therefore tan(x + 450) = sin(x + 450) / cos(x + 450) = cos(x) / - sin(x) = -cot(x). since sin and cos are defined for all real numbers and the problem is only with tan as it is not defined for certain real numbers((4n+1)90, (4n-1)90, -(4n+1)90, -(4n-1)90) tan(x + 450) cannot be simplified using tangent sum formula but can be simplified using sin and cos formulas.

We now attempt to differentiate between the trigonometric equation that is identity and the trigonometric equation that is not identity. We have from symbolic logic the definition of identity as x is said to be identical with y if x takes a value "u" implies y takes the value "u". But x is said to be equal to y if x takes a value "u" implies y takes value "u" and y takes value "u" implies x takes value "u". i.e. x is not identical with y if x takes a certain value and at the same time y does not take the same value.

We say that a trigonometric expression ex1 is identical with a trigonometric expression ex2 if whenever ex1 takes a value "p" ex2 also takes the value "p". We give an example of trigonometric identity.The trigonometric expression (1 - tan2(x)) / (1 + tan2(x)) is identical with the trigonometric expression cos2(x) - sin2(x). Whenever the first expression takes a definite value say, p, tan(x) is well defined and (1 - tan2(x)) / (1 + tan2(x)) can be simplified as (1- sin2(x) / cos2(x)) / (1+ sin2(x) / cos2(x)) i.e. cos2(x) - sin2(x) which also takes the value p.

But when cos2(x) - sin2(x) takes value -1 in a particular case when x = 90 then in that case (1 - tan2(x)) / (1 + tan2(x)) is undefined as tan 90 is undefined. That is (1 - tan2(x)) / (1 + tan2(x)) can be simplified to the other expression and the trigonometric equation (1 - tan2(x)) / (1 + tan2(x)) = cos2(x) - sin2(x) is an identity.Let us consider tan(x) = . This equation is not an identity as tan(x) takes any real value that does not imply that the right hand side which is constant also takes the same value and the right hand side which takes the value does not imply that the left hand side will take the value always.

In fact we need to solve this equation and find the values of x which would satisfy the equation.